The next term is 2^(2^(2^(2^16))) - 3, which is too large to display in the DATA lines.

Another version of the Ackermann numbers is the sequence 1^1, 2^^2, 3^^^3, 4^^^^4, 5^^^^^5, ..., which begins 1, 4, 3^3^3^... (where the number of 3's in the tower is 3^3^3 = 7625597484987), ... [Conway and Guy]. This grows too rapidly to have its own entry in the OEIS.

An even more rapidly growing sequence is the Conway-Guy sequence 1, 2->2, 3->3->3, 4->4->4->4, ..., which agrees with the sequence in the previous comment for n <= 3, but then the 4th term is very much larger than 4^^^^4.

A189896(n) = succ(0), 1+1, 2*2, 3^3,..., also called Ackermann numbers, is a weaker version of the above sequence.

The Ackermann functions are well-known to be simple examples of computable (implementable using a combination of while/for-loops) but not primitive recursive (implementable using only a FINITE number of do-while/for-loops) functions.

See A054871 for the definitions of the hyperoperations (a[n]b and H_n(a,b)).